CN110095983B - Mobile robot prediction tracking control method based on path parameterization - Google Patents

Mobile robot prediction tracking control method based on path parameterization Download PDF

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CN110095983B
CN110095983B CN201910321935.2A CN201910321935A CN110095983B CN 110095983 B CN110095983 B CN 110095983B CN 201910321935 A CN201910321935 A CN 201910321935A CN 110095983 B CN110095983 B CN 110095983B
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俞立
陈旭
吴锦辉
刘安东
仇翔
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Zhejiang University of Technology ZJUT
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Abstract

A mobile robot prediction tracking control method based on path parameterization comprises the following steps: 1) establishing a path tracking error model of the mobile robot; 2) defining a parameterized path update rule; 3) designing a performance index function; 4) defining a prediction model vector description; 5) and solving quadratic optimal control quantity by a Newton method. The invention provides a prediction tracking control method which can effectively solve the problem that the set speed and the actual speed of a mobile robot cannot be quickly matched.

Description

Mobile robot prediction tracking control method based on path parameterization
Technical Field
The invention relates to the field of path tracking control of a mobile robot, in particular to a model prediction control method based on path parameterization, which is provided because the actual speed and the set speed of the mobile robot cannot be quickly matched.
Background
With the development of software and hardware technologies and control technologies, robots have been widely used in various industries. The path tracking control technology of the mobile robot relates to knowledge achievement of multiple cross subjects such as mechanical engineering, electrical automation, sensing technology, computer technology, image processing technology and the like, and has gained high attention from the world in various fields such as civil use, industry and military. The mobile robot path tracking control technology is also suitable for other scenes, such as ship paths, lathe cutting paths, automatic driving and the like. Therefore, aiming at the research of the path tracking control technology of the mobile robot, the theoretical result of the motion control of the mobile robot can be enriched, the higher and higher requirements of multiple fields on the motion control technology can be met, and the method has great theoretical and engineering significance.
However, in an actual environment, especially in a complex working environment, various uncertain factors interfere with path tracking of the mobile robot, wherein in the operation process of the robot, there is a problem that an error becomes large because a real-time set speed and an actual speed cannot be matched quickly, which brings opportunities and challenges to mobile robot technology.
Compared with other control methods, the model prediction control method can correct uncertainty caused by model mismatch, interference and the like in time, has the advantages of convenience in modeling, stable system, good expansibility and the like, and is popular with scientific researchers. Cortex et al designs a multi-robot chain system based on a prediction model, divides the prediction model into six independent modules, respectively performs model verification on the single module, and respectively designs a controller, so that overshoot and stabilization time are effectively controlled. Karl Worthmann et al propose a model-based predictive control scheme for the steering problem of an incomplete mobile robot, establish a predictive model, strictly analyze stability, and verify the steering effect of the incomplete mobile robot. In order to successfully control two systems of residents in a paper (wheeled robot formation based on a predictive control method), the xiaozhen et al adopts Model Predictive Control (MPC) as a control method in an experiment. The model predictive control solves the optimal problem by constructing a Quadratic Programming (QP) with constraints, and iteratively solves the optimal problem in real time to obtain the optimal control input. Liuyang et al in the paper (Model Predictive Control based mobile robot path tracking Control) using Nonlinear Model Predictive Control (NMPC) has mechanisms of roll optimization and feedback correction, can handle the state constraints and input constraints of the system. However, these results do not take into account the actual speed matching problem of the mobile robot, and when the control amount is input after the processes of prediction model, rolling optimization and quadratic problem solving, the mobile robot needs to be accelerated or decelerated to reach a set value, and the process is affected by the problems of battery, motor, drive, inertia and the like, so that it is very necessary to study the matching problem between the input speed and the actual speed of the mobile robot.
Disclosure of Invention
In order to solve the problem that the prior art cannot solve the problem that the set speed and the actual speed of the control quantity of the mobile robot cannot be quickly matched, the invention provides a mobile robot prediction tracking control method based on path parameterization.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a mobile robot prediction tracking control method based on path parameterization comprises the following steps:
1) establishing a robot kinematic model, wherein x is [ x, y, theta ]]TFor the actual pose of the robot, (x, y) is the actual position of the robot, θ is the actual angle of the robot, and r is defined as [ x ═ xr,yr,θr]TFor the virtual machine reference pose, (x)r,yr) For the virtual machine reference position, θrFor the virtual robot reference angle, the robot kinematics model is:
Figure BDA0002035019490000021
Figure BDA0002035019490000022
wherein v is the actual linear velocity of the robot, omega is the actual angular velocity of the robot, vrFor the virtual machine reference line speed, omegarFor the robot reference angular velocity, the tracking error model is:
Figure BDA0002035019490000023
wherein, [ x ]e,ye,θe]Is an error vector, (x)e,ye) For deviations of the actual position from the reference position, θeIs an angular deviation;
2) establishing a linear error model of the mobile robot, and deriving the equation (3):
Figure BDA0002035019490000031
the state space equation is linearized at the equilibrium point according to equation (4) as follows:
Figure BDA0002035019490000032
wherein,
Figure BDA0002035019490000033
in order to be a state error vector,
Figure BDA0002035019490000034
inputting deviation vectors, matrices, for robot control
Figure BDA0002035019490000035
Matrix array
Figure BDA0002035019490000036
Discretizing equation (5) to obtain:
Figure BDA0002035019490000037
wherein k is the sampling time,
Figure BDA0002035019490000038
for the state error vector of the robot at time k,
Figure BDA0002035019490000039
the deviation amount is input for the control of time k,
Figure BDA00020350194900000310
Figure BDA00020350194900000311
Tsis a sampling period;
3) defining a parameterized expected path:
P＝{r(k)∈Rn|r(k)＝p(θr(k))} (7)
where P is the parameterized expected path, r (k) is the reference position at time k, P (θ)r(k) A path at time k, θr(k) Is the path parameter at time k, θr(k) The parameter updating method comprises the following steps:
Figure BDA00020350194900000312
wherein, ω isp(k) The angular velocity is expected for the path at time k,
Figure BDA00020350194900000313
for a linear expression relating time k to the control input offset, the relationship is as follows:
Figure BDA00020350194900000314
where λ is a gain scalar, C ═ C1 c2]Is a gain matrix related to the control input error vector;
4) the following predicted performance indicators are defined:
Figure BDA00020350194900000315
wherein,
Figure BDA00020350194900000316
is a state deviation penalty term, Q is a state weighting matrix,
Figure BDA00020350194900000317
indicating the predicted value of the state at time k versus time k + i,
Figure BDA0002035019490000041
is a penalty term of the reference control quantity and the path expected control quantity at the moment k, vr(k + i | k) is the predicted value of the reference linear velocity at time k + i, ωr(k + i | k) is the predicted value of the reference angular velocity at the time k + i, vp(k + i | k) is the predicted desired linear velocity of the path at time k + i, ωp(k + i | k) is the predicted value of the desired angular velocity of the path at time k + i, N is the predicted time domain of the state deviation,
Figure BDA0002035019490000042
is a control input deviation penalty term, R is an input weighting matrix,
Figure BDA0002035019490000043
a predicted value representing a control input deviation amount at time k to time k + i, where M is a predicted time domain of the control input deviation amount;
5) defining a prediction model vector description, and obtaining a prediction model according to the formula (6) as follows:
Figure BDA0002035019490000044
wherein,
Figure BDA0002035019490000045
is the error state prediction vector and is,
Figure BDA0002035019490000046
is to control the input offset prediction vector to,
Figure BDA0002035019490000047
is obtained from the formula (8)
Figure BDA0002035019490000048
The prediction model for Δ g (k + i | k) is then:
Figure BDA0002035019490000049
wherein g (k) ═ Δ g (k) … Δ g (k + M-1)]Is a gain matrix, RrIs omegarThe curvature radius corresponding to the moment is the optimized performance index:
Figure BDA00020350194900000410
6) order to
Figure BDA00020350194900000411
And defining quadratic programming problem description according to equations (11), (12) and (13):
Figure BDA00020350194900000412
wherein D ═ HTQH+GTG+R，ET＝(Fx(k))TQH，d＝(Fx(k))TQfx (k), the Newton method iterative formula is:
Figure BDA0002035019490000051
wherein,
Figure BDA0002035019490000052
d is called a hessian matrix and,
Figure BDA0002035019490000053
according to
Figure BDA0002035019490000054
Calculation of
Figure BDA0002035019490000055
And successively backward-pushed, and the following are obtained from the quadratic termination in Newton's method:
Figure BDA0002035019490000056
the minimum point of the quadratic programming problem description formula (14) is
Figure BDA0002035019490000057
And is
Figure BDA0002035019490000058
The first term in (1) calculates the control input quantity at the current k moment
Figure BDA0002035019490000059
The technical conception of the invention is as follows: firstly, a dynamic model of the mobile robot under a linear system is established, and an update equation of a parameterized expected path is given. Then, a prediction performance index function is defined, and prediction model vector description is deduced by combining a state space equation. And then, solving the optimal control quantity based on Newton method quadratic programming. And finally, analyzing the feasibility of the algorithm through a simulation experiment, and designing an experimental platform of the path tracking control system of the mobile robot to verify the actual significance of the algorithm in order to explain the performance of the method.
The invention has the following beneficial effects: the method comprises the steps of defining a parameterized path to be related to the actual robot state by establishing a kinematic error model, calculating the optimal control quantity by combining a model prediction control algorithm and Newton method quadratic programming, and solving the problem that the input speed of the control quantity and the actual speed of the robot cannot be matched quickly in the path tracking process of the mobile robot.
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FIG. 1 is a schematic diagram of mobile robot error model coordinate establishment.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1, a mobile robot predictive tracking control method based on path parameterization includes the following steps:
1) establishing a robot kinematic model, wherein x is [ x, y, theta ]]TFor the actual pose of the robot, (x, y) is the actual position of the robot, θ is the actual angle of the robot, and r is defined as [ x ═ xr,yr,θr]TFor the virtual machine reference pose, (x)r,yr) For the virtual machine reference position, θrFor the virtual robot reference angle, the robot kinematics model is:
Figure BDA0002035019490000061
Figure BDA0002035019490000062
wherein v is the actual linear velocity of the robot, omega is the actual angular velocity of the robot, vrFor the virtual machine reference line speed, omegarFor the robot reference angular velocity, the tracking error model is:
Figure BDA0002035019490000063
wherein, [ x ]e,ye,θe]Is an error vector, (x)e,ye) For deviations of the actual position from the reference position, θeIs an angular deviation;
2) establishing a linear error model of the mobile robot, and deriving the equation (3):
Figure BDA0002035019490000064
the state space equation is linearized at the equilibrium point according to equation (4) as follows:
Figure BDA0002035019490000065
wherein,
Figure BDA0002035019490000066
in order to be a state error vector,
Figure BDA0002035019490000067
inputting deviation vectors, matrices, for robot control
Figure BDA0002035019490000068
Matrix array
Figure BDA0002035019490000069
Discretizing equation (5) to obtain:
Figure BDA00020350194900000610
wherein k is the sampling time,
Figure BDA00020350194900000611
for the state error vector of the robot at time k,
Figure BDA00020350194900000612
the deviation amount is input for the control of time k,
Figure BDA00020350194900000613
Figure BDA00020350194900000614
Tsis a sampling period;
3) defining a parameterized expected path:
P＝{r(k)∈Rn|r(k)＝p(θr(k))} (7)
where P is the parameterized expected path, r (k) is the reference position at time k, P (θ)r(k) A path at time k, θr(k) Is the path parameter at time k, θr(k) The parameter updating method comprises the following steps:
Figure BDA0002035019490000071
wherein, ω isp(k) The angular velocity is expected for the path at time k,
Figure BDA0002035019490000072
for a linear expression relating time k to the control input offset, the relationship is as follows:
Figure BDA0002035019490000073
where λ is a gain scalar, C ═ C1 c2]Is a gain matrix related to the control input error vector;
4) the following predicted performance indicators are defined:
Figure BDA0002035019490000074
wherein,
Figure BDA0002035019490000075
is a state deviation penalty term, Q is a state weighting matrix,
Figure BDA0002035019490000076
indicating the predicted value of the state at time k versus time k + i,
Figure BDA0002035019490000077
is a penalty term of the reference control quantity and the path expected control quantity at the moment k, vr(k + i | k) is the predicted value of the reference linear velocity at time k + i, ωr(k + i | k) is the predicted value of the reference angular velocity at the time k + i, vp(k + i | k) is the predicted desired linear velocity of the path at time k + i, ωp(k + i | k) is the predicted value of the desired angular velocity of the path at time k + i, N is the predicted time domain of the state deviation,
Figure BDA0002035019490000078
is a control input deviation penalty term, R is an input weighting matrix,
Figure BDA0002035019490000079
a predicted value representing a control input deviation amount at time k to time k + i, where M is a predicted time domain of the control input deviation amount;
5) defining a prediction model vector description, and obtaining a prediction model according to the formula (6) as follows:
Figure BDA00020350194900000710
wherein,
Figure BDA00020350194900000711
is the error state prediction vector and is,
Figure BDA00020350194900000712
is to control the input offset prediction vector to,
Figure BDA0002035019490000081
is obtained from the formula (8)
Figure BDA0002035019490000082
The prediction model for Δ g (k + i | k) is then:
Figure BDA0002035019490000083
wherein g (k) ═ Δ g (k) … Δ g (k + M-1)]Is a gain matrix, RrIs omegarThe curvature radius corresponding to the moment is the optimized performance index:
Figure BDA0002035019490000084
6) order to
Figure BDA0002035019490000085
And defining quadratic programming problem description according to equations (11), (12) and (13):
Figure BDA0002035019490000086
wherein D ═ HTQH+GTG+R，ET＝(Fx(k))TQH，d＝(Fx(k))TQfx (k), the Newton method iterative formula is:
Figure BDA0002035019490000087
wherein,
Figure BDA0002035019490000088
d is called a hessian matrix and,
Figure BDA0002035019490000089
according to
Figure BDA00020350194900000810
Calculation of
Figure BDA00020350194900000811
And successively backward-pushed, and the following are obtained from the quadratic termination in Newton's method:
Figure BDA00020350194900000812
the minimum point of the quadratic programming problem description formula (14) is
Figure BDA00020350194900000813
And is
Figure BDA00020350194900000814
The first term in (1) calculates the control input quantity at the current k moment
Figure BDA00020350194900000815

Claims (1)

1. A mobile robot prediction tracking control method based on path parameterization is characterized by comprising the following steps:
1) establishing a robot kinematic model, wherein x is [ x, y, theta ]]TFor the actual pose of the robot, (x, y) is the actual position of the robot, θ is the actual angle of the robot, and r is defined as [ x ═ xr,yr,θr]TFor the virtual machine reference pose, (x)r,yr) For the virtual machine reference position, θrFor the virtual robot reference angle, the robot kinematics model is:
Figure FDA0002035019480000011
Figure FDA0002035019480000012
wherein v is the actual linear velocity of the robot, omega is the actual angular velocity of the robot, vrFor the virtual machine reference line speed, omegarFor the robot reference angular velocity, the tracking error model is:
Figure FDA0002035019480000013
wherein, [ x ]e,ye,θe]Is an error vector, (x)e,ye) For deviations of the actual position from the reference position, θeIs an angular deviation;
2) establishing a linear error model of the mobile robot, and deriving the equation (3):
Figure FDA0002035019480000014
the state space equation is linearized at the equilibrium point according to equation (4) as follows:
Figure FDA0002035019480000015
wherein,
Figure FDA0002035019480000016
in order to be a state error vector,
Figure FDA0002035019480000017
inputting deviation vectors, matrices, for robot control
Figure FDA0002035019480000018
Matrix array
Figure FDA0002035019480000019
Discretizing equation (5) to obtain:
Figure FDA00020350194800000110
wherein k is the sampling time,
Figure FDA00020350194800000111
for the state error vector of the robot at time k,
Figure FDA00020350194800000112
the deviation amount is input for the control of time k,
Figure FDA00020350194800000113
Figure FDA00020350194800000114
Tsis a sampling period;
3) defining a parameterized expected path:
P＝{r(k)∈Rn|r(k)＝p(θr(k))} (7)
where P is the parameterized expected path, r (k) is the reference position at time k, P (θ)r(k) A path at time k, θr(k) Is the path parameter at time k, θr(k) The parameter updating method comprises the following steps:
Figure FDA0002035019480000021
wherein, ω isp(k) The angular velocity is expected for the path at time k,
Figure FDA0002035019480000022
for a linear expression relating time k to the control input offset, the relationship is as follows:
Figure FDA0002035019480000023
where λ is a gain scalar, C ═ C1 c2]Is a gain matrix related to the control input error vector;
4) the following predicted performance indicators are defined:
Figure FDA0002035019480000024
wherein,
Figure FDA0002035019480000025
is a state deviation penalty term, Q is a state weighting matrix,
Figure FDA0002035019480000026
indicating the predicted value of the state at time k versus time k + i,
Figure FDA0002035019480000027
is a penalty term of the reference control quantity and the path expected control quantity at the moment k, vr(k + i | k) is the predicted value of the reference linear velocity at time k + i, ωr(k + i | k) is the predicted value of the reference angular velocity at the time k + i, vp(k + i | k) is the predicted desired linear velocity of the path at time k + i, ωp(k + i | k) is the predicted value of the desired angular velocity of the path at time k + i, N is the predicted time domain of the state deviation,
Figure FDA0002035019480000028
is a control input deviation penalty term, R is an input weighting matrix,
Figure FDA0002035019480000029
a predicted value representing a control input deviation amount at time k to time k + i, where M is a predicted time domain of the control input deviation amount;
5) defining a prediction model vector description, and obtaining a prediction model according to the formula (6) as follows:
Figure FDA00020350194800000210
wherein,
Figure FDA00020350194800000211
is the error state prediction vector and is,
Figure FDA00020350194800000212
is to control the input offset prediction vector to,
Figure FDA00020350194800000213
is obtained from the formula (8)
Figure FDA00020350194800000214
The prediction model for Δ g (k + i | k) is then:
Figure FDA00020350194800000215
wherein g (k) ═ Δ g (k) … Δ g (k + M-1)]Is a gain matrix, RrIs omegarThe curvature radius corresponding to the moment is the optimized performance index:
Figure FDA0002035019480000031
6) order to
Figure FDA0002035019480000032
And defining quadratic programming problem description according to equations (11), (12) and (13):
Figure FDA0002035019480000033
wherein D ═ HTQH+GTG+R，ET＝(Fx(k))TQH，d＝(Fx(k))TQfx (k), the Newton method iterative formula is:
Figure FDA0002035019480000034
wherein,
Figure FDA0002035019480000035
d is called a hessian matrix and,
Figure FDA0002035019480000036
according to
Figure FDA0002035019480000037
Calculation of
Figure FDA0002035019480000038
And successively backward-pushed, and the following are obtained from the quadratic termination in Newton's method:
Figure FDA0002035019480000039
the minimum point of the quadratic programming problem description formula (14) is
Figure FDA00020350194800000310
And is
Figure FDA00020350194800000311
The first term in (1) calculates the control input quantity at the current k moment
Figure FDA00020350194800000312
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